计算cosπ/8的四次方+cos3π/8的四次方+5π/8的四次方+cos7π/8的四次方之和

题目

计算cosπ/8的四次方+cos3π/8的四次方+5π/8的四次方+cos7π/8的四次方之和

答案

cos7π/8
=-cos(π-7π/8)
=-cosπ/8
同理cos5π/8=-cos3π/8
又cos3π/8=sin(π/2-3π/8)=sinπ/8
所以cosπ/8的四次方+cos3π/8的四次方+5π/8的四次方+cos7π/8的四次方
=2(cosπ/8的四次方+cos3π/8的四次方)
=2(cosπ/8的四次方+sinπ/8的四次方)
=2[(cosπ/8的二次方+sinπ/8的二次方)^2-2cosπ/8的二次方*sinπ/8的二次方]
=2[1-(sinπ/4)^2/2]
=2*(1-1/4)
=3/2